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Properties

Label 2-72-72.59-c3-0-16
Degree 22
Conductor 7272
Sign 0.998−0.0552i0.998 - 0.0552i
Analytic cond. 4.248134.24813
Root an. cond. 2.061102.06110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.80 + 2.17i)2-s + (−3.91 + 3.42i)3-s + (−1.46 − 7.86i)4-s + (7.74 − 13.4i)5-s + (−0.371 − 14.6i)6-s + (−4.08 + 2.35i)7-s + (19.7 + 11.0i)8-s + (3.59 − 26.7i)9-s + (15.1 + 41.1i)10-s + (4.32 − 2.49i)11-s + (32.6 + 25.7i)12-s + (66.4 + 38.3i)13-s + (2.25 − 13.1i)14-s + (15.6 + 79.0i)15-s + (−59.7 + 23.0i)16-s − 120. i·17-s + ⋯
L(s)  = 1  + (−0.639 + 0.769i)2-s + (−0.752 + 0.658i)3-s + (−0.182 − 0.983i)4-s + (0.693 − 1.20i)5-s + (−0.0252 − 0.999i)6-s + (−0.220 + 0.127i)7-s + (0.873 + 0.487i)8-s + (0.132 − 0.991i)9-s + (0.480 + 1.30i)10-s + (0.118 − 0.0685i)11-s + (0.784 + 0.619i)12-s + (1.41 + 0.818i)13-s + (0.0430 − 0.250i)14-s + (0.268 + 1.36i)15-s + (−0.933 + 0.359i)16-s − 1.72i·17-s + ⋯

Functional equation

Λ(s)=(72s/2 ΓC(s) L(s)=((0.998−0.0552i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0552i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(72s/2 ΓC(s+3/2) L(s)=((0.998−0.0552i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0552i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7272    =    23⋅322^{3} \cdot 3^{2}
Sign: 0.998−0.0552i0.998 - 0.0552i
Analytic conductor: 4.248134.24813
Root analytic conductor: 2.061102.06110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ72(59,⋅)\chi_{72} (59, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 72, ( :3/2), 0.998−0.0552i)(2,\ 72,\ (\ :3/2),\ 0.998 - 0.0552i)

Particular Values

L(2)L(2) ≈\approx 0.902292+0.0249448i0.902292 + 0.0249448i
L(12)L(\frac12) ≈\approx 0.902292+0.0249448i0.902292 + 0.0249448i
L(52)L(\frac{5}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(1.80−2.17i)T 1 + (1.80 - 2.17i)T
3 1+(3.91−3.42i)T 1 + (3.91 - 3.42i)T
good5 1+(−7.74+13.4i)T+(−62.5−108.i)T2 1 + (-7.74 + 13.4i)T + (-62.5 - 108. i)T^{2}
7 1+(4.08−2.35i)T+(171.5−297.i)T2 1 + (4.08 - 2.35i)T + (171.5 - 297. i)T^{2}
11 1+(−4.32+2.49i)T+(665.5−1.15e3i)T2 1 + (-4.32 + 2.49i)T + (665.5 - 1.15e3i)T^{2}
13 1+(−66.4−38.3i)T+(1.09e3+1.90e3i)T2 1 + (-66.4 - 38.3i)T + (1.09e3 + 1.90e3i)T^{2}
17 1+120.iT−4.91e3T2 1 + 120. iT - 4.91e3T^{2}
19 1−30.8T+6.85e3T2 1 - 30.8T + 6.85e3T^{2}
23 1+(−91.9+159.i)T+(−6.08e3−1.05e4i)T2 1 + (-91.9 + 159. i)T + (-6.08e3 - 1.05e4i)T^{2}
29 1+(−28.8−49.9i)T+(−1.21e4+2.11e4i)T2 1 + (-28.8 - 49.9i)T + (-1.21e4 + 2.11e4i)T^{2}
31 1+(37.8+21.8i)T+(1.48e4+2.57e4i)T2 1 + (37.8 + 21.8i)T + (1.48e4 + 2.57e4i)T^{2}
37 1+108.iT−5.06e4T2 1 + 108. iT - 5.06e4T^{2}
41 1+(230.+133.i)T+(3.44e4+5.96e4i)T2 1 + (230. + 133. i)T + (3.44e4 + 5.96e4i)T^{2}
43 1+(105.+181.i)T+(−3.97e4+6.88e4i)T2 1 + (105. + 181. i)T + (-3.97e4 + 6.88e4i)T^{2}
47 1+(−191.−332.i)T+(−5.19e4+8.99e4i)T2 1 + (-191. - 332. i)T + (-5.19e4 + 8.99e4i)T^{2}
53 1−419.T+1.48e5T2 1 - 419.T + 1.48e5T^{2}
59 1+(17.1+9.91i)T+(1.02e5+1.77e5i)T2 1 + (17.1 + 9.91i)T + (1.02e5 + 1.77e5i)T^{2}
61 1+(−158.+91.4i)T+(1.13e5−1.96e5i)T2 1 + (-158. + 91.4i)T + (1.13e5 - 1.96e5i)T^{2}
67 1+(215.−373.i)T+(−1.50e5−2.60e5i)T2 1 + (215. - 373. i)T + (-1.50e5 - 2.60e5i)T^{2}
71 1+772.T+3.57e5T2 1 + 772.T + 3.57e5T^{2}
73 1+433.T+3.89e5T2 1 + 433.T + 3.89e5T^{2}
79 1+(−644.+372.i)T+(2.46e5−4.26e5i)T2 1 + (-644. + 372. i)T + (2.46e5 - 4.26e5i)T^{2}
83 1+(408.−235.i)T+(2.85e5−4.95e5i)T2 1 + (408. - 235. i)T + (2.85e5 - 4.95e5i)T^{2}
89 1+206.iT−7.04e5T2 1 + 206. iT - 7.04e5T^{2}
97 1+(−713.−1.23e3i)T+(−4.56e5+7.90e5i)T2 1 + (-713. - 1.23e3i)T + (-4.56e5 + 7.90e5i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−14.27759214656699717062522777429, −13.22347656165244026723334348791, −11.74730571715629264487219767236, −10.53690067021129966330984352147, −9.220936306144582194223931772680, −8.894460087751121125891410633770, −6.77909355345201367566808049869, −5.63777184837284400118144680563, −4.63481230646425496867790029900, −0.920384015480600605360090159118, 1.53524323135448398526695052450, 3.35122073978563504164958518563, 5.90581879505814656616080057838, 7.01583587799939749787375844818, 8.338657996349754676127543698095, 10.09691073964804319866231829844, 10.73761277252149698920135828252, 11.62180636349010140816520465424, 13.07064209235846150379263086051, 13.55295952558952452701592322398

Graph of the ZZ-function along the critical line